{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Differentiation formulas for the \+ Fourier transform" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nana imo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Differentiation formulas for the \+ Fourier transform" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "1st differentiation formula" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega)" "6#/*&%#Fr G\"\"\"7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 11 ", and that " } {XPPEDIT 18 0 "f(x) -> 0" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG 6\"\"\"!F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x -> infinity" "6#f *6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrow G6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F r*[`f '`(x)]=i*omega*F(omega)" "6#/*&%#FrG\"\"\"7#-%$f~'G6#%\"xGF&*(% \"iGF&%&omegaGF&-%\"FG6#F.F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 11 "___________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "If f '(" }{TEXT 264 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``= g(x)" "6#/%!G-%\"gG6#%\" xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Fr*[`f '`(x)] = G(omega)" "6#/ *&%#FrG\"\"\"7#-%$f~'G6#%\"xGF&-%\"GG6#%&omegaG" }{TEXT -1 6 ", then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G(omega) = Int(g(x) *exp(-i*omega*x),x=-infinity..infinity)" "6#/-%\"GG6#%&omegaG-%$IntG6$ *&-%\"gG6#%\"xG\"\"\"-%$expG6#,$*(%\"iGF0F'F0F/F0!\"\"F0/F/;,$%)infini tyGF7F;" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(g(x)*exp(-i*omega*x),x) = Int( u*``(dv/dx),x);" "6#/-%$IntG6$*&-%\"gG6#%\"xG\"\"\"-%$expG6#,$*(%\"iGF ,%&omegaGF,F+F,!\"\"F,F+-F%6$*&%\"uGF,-%!G6#*&%#dvGF,%#dxGF4F,F+" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = exp(-i*omega*x);" "6#/%\"uG- %$expG6#,$*(%\"iG\"\"\"%&omegaGF+%\"xGF+!\"\"" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "v=f(x)" "6#/%\"vG-%\"fG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "Hence, u sing integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x ) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF) %#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0 " }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "G(omega)= Limit(``,R=infinity)" "6#/-%\"GG6#%&omegaG-%& LimitG6$%!G/%\"RG%)infinityG" }{XPPEDIT 18 0 "f(x)*exp(-i*omega*x);" " 6#*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*(%\"iGF(%&omegaGF(F'F(!\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R , ``],[ -R, ``])" "6#-%*PI ECEWISEG6$7$%\"RG%!G7$,$F'!\"\"F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-I nt(f(x)*``(-i*omega*exp(-i*omega*x)),x = -infinity .. infinity);" "6#, $-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%!G6#,$*(%\"iGF,%&omegaGF,-%$expG6#,$* (F2F,F3F,F+F,!\"\"F,F9F,/F+;,$%)infinityGF9F=F9" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 0+i*omega*Int(f( x)*exp(-i*omega*x),x = -infinity .. infinity);" "6#/%!G,&\"\"!\"\"\"*( %\"iGF'%&omegaGF'-%$IntG6$*&-%\"fG6#%\"xGF'-%$expG6#,$*(F)F'F*F'F2F'! \"\"F'/F2;,$%)infinityGF8F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "2nd differentiation formula" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "S uppose that " }{XPPEDIT 18 0 "Fr*[f(x)] = F(omega);" "6#/*&%#FrG\"\"\" 7#-%\"fG6#%\"xGF&-%\"FG6#%&omegaG" }{TEXT -1 12 " , and that " }{TEXT 262 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) -> 0" "6#f*6#-%\"fG6#% \"xG7\"6$%)operatorG%&arrowG6\"\"\"!F-F-F-" }{TEXT -1 4 " as " } {XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"% )infinityGF*F*F*" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x -> -infinity" " 6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[x*f(x)] = i;" "6#/*&%#FrG\"\"\"7# *&%\"xGF&-%\"fG6#F)F&F&%\"iG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/(d*ome ga)" "6#*&%\"dG\"\"\"*&F$F%%&omegaGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F(omega);" "6#-%\"FG6#%&omegaG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 13 "_____________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "d/(d* omega)" "6#*&%\"dG\"\"\"*&F$F%%&omegaGF%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "F(omega)= d/(d*omega)" "6#/-%\"FG6#%&omegaG*&%\"dG\"\" \"*&F)F*F'F*!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*exp(-i*ome ga*x),x = -infinity .. infinity);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-% $expG6#,$*(%\"iGF+%&omegaGF+F*F+!\"\"F+/F*;,$%)infinityGF3F7" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(d iff([f(x)*exp(-i*omega*x)],omega),x = -infinity .. infinity);" "6#/%!G -%$IntG6$-%%diffG6$7#*&-%\"fG6#%\"xG\"\"\"-%$expG6#,$*(%\"iGF1%&omegaG F1F0F1!\"\"F1F8/F0;,$%)infinityGF9F=" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(-i*x*f(x)*exp(-i*omega*x) ,x = -infinity .. infinity);" "6#/%!G-%$IntG6$,$**%\"iG\"\"\"%\"xGF+-% \"fG6#F,F+-%$expG6#,$*(F*F+%&omegaGF+F,F+!\"\"F+F6/F,;,$%)infinityGF6F :" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = -i*Int(x*f(x)*exp(-i*omega*x),x = -infinity .. infinity)" "6#/%!G ,$*&%\"iG\"\"\"-%$IntG6$*(%\"xGF(-%\"fG6#F-F(-%$expG6#,$*(F'F(%&omegaG F(F-F(!\"\"F(/F-;,$%)infinityGF7F;F(F7" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -i*Fr*[x*f(x)];" "6#/%!G,$* (%\"iG\"\"\"%#FrGF(7#*&%\"xGF(-%\"fG6#F,F(F(!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr* [1/(a^2+x^2)] = Pi/a;" "6#/*&%#FrG\"\"\"7#*&F&F&,&*$%\"aG\"\"#F&*$%\"x GF,F&!\"\"F&*&%#PiGF&F+F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-a*abs( omega))" "6#-%$expG6#,$*&%\"aG\"\"\"-%$absG6#%&omegaGF)!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 265 1 "a" } {TEXT -1 14 " is positive. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[1/(a^2+x^2)] = -2*x/(a^2+x^2)^2" "6#/7#*&\"\"\"F&,&*$% \"aG\"\"#F&*$%\"xGF*F&!\"\",$*(F*F&F,F&*$,&*$F)F*F&*$F,F*F&F*F-F-" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 39 "so, by the1st different iation formula, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F r*[-2*x/((a^2+x^2)^2)] = i*omega;" "6#/*&%#FrG\"\"\"7#,$*(\"\"#F&%\"xG F&*$,&*$%\"aGF*F&*$F+F*F&F*!\"\"F1F&*&%\"iGF&%&omegaGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/a;" "6#*&%#PiG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-a*abs(omega));" "6#-%$expG6#,$*&%\"aG\"\"\"-%$abs G6#%&omegaGF)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Henc e" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[x/((a^2+x^2) ^2)] = -i*Pi*omega/(2*a);" "6#/*&%#FrG\"\"\"7#*&%\"xGF&*$,&*$%\"aG\"\" #F&*$F)F.F&F.!\"\"F&,$**%\"iGF&%#PiGF&%&omegaGF&*&F.F&F-F&F0F0" } {TEXT -1 2 " " }{XPPEDIT 18 0 "exp(-a*abs(omega));" "6#-%$expG6#,$*&% \"aG\"\"\"-%$absG6#%&omegaGF)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "interface(s howassumed=0): assume(a>0):\nx/(a^2+x^2)^2;\n`Fourier transform`=inttr ans[fourier](%,x,omega);\n``=convert(rhs(%),piecewise,omega);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&*$)%#a|irG\"\"#F%F%*$)F $F*F%F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG*, ^##!\"\"\"\"#\"\"\"%#PiGF*%&omegaGF*,&*&-%$expG6#*&%#a|irGF*F,F*F*-%\" HG6#,$F,F(F*F**&-F06#,$F2F(F*-F56#F,F*F*F*F3F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%*PIECEWISEG6%7$*,^##!\"\"\"\"#\"\"\"%#a|irGF,-%$e xpG6#*&F/F.%&omegaGF.F.%#PiGF.F4F.2F4\"\"!7$*(^#%*undefinedGF.F5F.F/F, /F4F77$*,F*F.F/F,-F16#,$F3F,F.F5F.F4F.2F7F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "In particular, when " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG \"\"\"" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Fr*[x/((1+x^2)^2)] = -i*Pi*omega/2;" "6#/*&%#FrG\"\"\"7 #*&%\"xGF&*$,&F&F&*$F)\"\"#F&F-!\"\"F&,$**%\"iGF&%#PiGF&%&omegaGF&F-F. F." }{TEXT -1 2 " " }{XPPEDIT 18 0 "exp(-abs(omega));" "6#-%$expG6#,$ -%$absG6#%&omegaG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> x/(1+x^2)^2:\n 'f(x)'=f(x);\nplot(f(x),x=-5..5,color=red,thickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\",&*$)F'\"\"# F)F)F)F)!\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 600 197 197 {PLOTDATA 2 "6( -%'CURVESG6#7ao7$$!\"&\"\"!$!3=8?9/(\\kR(!#?7$$!3YLLLe%G?y%!#<$!3]$yO3 beVR)F-7$$!3OmmT&esBf%F1$!3MgJlH>J6%*F-7$$!3ALL$3s%3zVF1$!3Q\"ROHv8d2 \"!#>7$$!3_LL$e/$QkTF1$!3qyW\\!z2yB\"F>7$$!3ommT5=q]RF1$!3oCJ0+2JK9F>7 $$!3ILL3_>f_PF1$!3sswD1Su\\;F>7$$!3K++vo1YZNF1$!3o=.ujnMA>F>7$$!3;LL3- OJNLF1$!3g1_V]`$*oAF>7$$!3p***\\P*o%Q7$F1$!3ol\\(RJP*)p#F>7$$!3Kmmm\"R Fj!HF1$!3'*\\>_o@scKF>7$$!33LL$e4OZr#F1$!3GlJ6-rBvQF>7$$!3u*****\\n\\! *\\#F1$!3#p%3>.;ngZF>7$$!3%)*****\\ixCG#F1$!3=`WE)=B#>fF>7$$!3#****** \\KqP2#F1$!3YW$**)>]8\"Q(F>7$$!39LL3-TC%)=F1$!3zivl#)e-+\"*F>7$$!3[mmm \"4z)e;F1$!3?i*)))\\bYy6!#=7$$!3Mmmmm`'zY\"F1$!3Ka:1M\"=[Z\"Fjp7$$!3#* ***\\(=t)eC\"F1$!3/(3Z'H#eE\">Fjp7$$!3!ommmh5$\\5F1$!3KLxVz_+xBFjp7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 36 "By the 2nd differentiation formula, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[x*exp(-a*x^2)] = i" "6#/*&%#FrG\" \"\"7#*&%\"xGF&-%$expG6#,$*&%\"aGF&*$F)\"\"#F&!\"\"F&F&%\"iG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/(d*omega)" "6#*&%\"dG\"\"\"*&F$F%%&omegaGF% !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sqrt(Pi/a)*exp(-omega^2/(4*a)) ]" "6#7#*&-%%sqrtG6#*&%#PiG\"\"\"%\"aG!\"\"F*-%$expG6#,$*&%&omegaG\"\" #*&\"\"%F*F+F*F,F,F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (-i*omega*sq rt(Pi)/(2*a*sqrt(a)))*exp(-omega^2/(4*a));" "6#/%!G*&,$**%\"iG\"\"\"%& omegaGF)-%%sqrtG6#%#PiGF)*(\"\"#F)%\"aGF)-F,6#F1F)!\"\"F4F)-%$expG6#,$ *&F*F0*&\"\"%F)F1F)F4F4F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be the function defined by \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE( [a^2-x^2, abs(x)=a])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&* $%\"aG\"\"#\"\"\"*$F'F/!\"\"2-%$absG6#F'F.7$\"\"!1F.F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 267 1 "a" }{TEXT -1 14 " is positive. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 14 " for the case " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\" \"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "f := x -> piecewise(-21DBF/F+7$$!3kmmmw))yr@F/F +7$$!3SLLL3V^&3#F/F+7$$!3;+++S(R#**>F/$\"3cOF/ $\"3qKv'po%>>M!#=7$$!30++++@)f#=F/$\"3g)*ez/P*yl'Ffn7$$!3/+++!=*\\UF/7$$!3XLLLtK5F8F/$\"3n^z!>!pzQAF/7$$!3_LLL$yP2D\"F/$\"3I >CM(*\\lNCF/7$$!3eLLL$HsV<\"F/$\"3v15l;(\\3i#F/7$$!3+-++]&)4n**Ffn$\"3 ^ye%\\Ypl+$F/7$$!37PLLL\\[%R)Ffn$\"3-'RSqAE`H$F/7$$!3G)*****\\&y!pmFfn $\"39*HRH\"RBbNF/7$$!3Y******\\O3E]Ffn$\"3iE?VJ[QZPF/7$$!3NKLLL3z6LFfn $\"3C$\\iZT?.*QF/7$$!3/LLLeGmCDFfn$\"3a@v^u2EORF/7$$!3sLLL$)[`PHD(4)pRF/7$$!3m0++]-6&)))!#>$\"3[MXe\"[0@*RF/7$$!3gSnmmmr[RFV$\"3 ;'oOwS%)***RF/7$$\"3S6LL3Uh9SFjs$\"3uCeF(G)Q)*RF/7$$\"3'oHLL3+TU)Fjs$ \"3M)fyPX.H*RF/7$$\"3AGL$efeLG\"Ffn$\"3+2]92*HN)RF/7$$\"3yELL$=2Vs\"Ff n$\"37^]PZwEqRF/7$$\"3Khmmm7+#\\#Ffn$\"3R<$poH**y$RF/7$$\"3)e*****\\`p fKFfn$\"3%Q)=DiQu$*QF/7$$\"36HLLLm&z\"\\Ffn$\"3=Fl_Dq8ePF/7$$\"3>(**** **z-6j'Ffn$\"3+A$elv%GgNF/7$$\"3q\"******4#32$)Ffn$\"35(fQ)pQ#*4LF/7$$ \"3q#****\\qM8F/$\"3,m .M?3d=AF/7$$\"3(HLLL5r5U\"F/$\"3w$)4F>pb!)>F/7$$\"3%)*******HSu]\"F/$ \"3S5f$>uBws\"F/7$$\"3Smm;HOq&e\"F/$\"3AjwW+Sa&[\"F/7$$\"3'HLL$ep'Rm\" F/$\"3.;\\dhR@J7F/7$$\"3Umm;\\%H&\\4N=F/$\"3Un.ZF/$\"3k^:n]Ey/LFfn7$$\"3#emm;@2 h*>F/$\"3iWGB`**fb:Fjs7$$\"3VKL$3jXr,#F/F+7$$\"3.******\\S=Q?F/F+7$$\" 3ilm;pCAf?F/F+7$$\"3mKLL))3E!3#F/F+7$$\"3JmmmExLA@F/F+7$$\"3]*****\\c9 W;#F/F+7$$\"3Lmmmmd'*GBF/F+7$$\"3j*****\\iN7]#F/F+7$$\"3aLLLt>:nEF/F+7 $$\"35LLL.a#o$GF/F+7$$\"3ammm^Q40IF/F+7$$\"3y******z]rfJF/F+7$$\"3gmmm c%GpL$F/F+7$$\"3/LLL8-V&\\$F/F+7$$\"3=+++XhUkOF/F+7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 "f(" }{TEXT 263 1 "x" }{TEXT -1 15 ") has t he form " }{XPPEDIT 18 0 "f(x)=(a^2-x^2)*(H(x+a)-H(x-a))" "6#/-%\"fG6# %\"xG*&,&*$%\"aG\"\"#\"\"\"*$F'F,!\"\"F-,&-%\"HG6#,&F'F-F+F-F--F26#,&F 'F-F+F/F/F-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 " H(x" "6#-%\"HG6#% \"xG" }{TEXT -1 36 " is the Heaviside function given by " }{XPPEDIT 18 0 "H(x)=PIECEWISE([0 , x<0],[1 , x>0" "6#/-%\"HG6#%\"xG-%*PIECEWISE G6$7$\"\"!2F'F,7$\"\"\"2F,F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[H(x+a)-H(x-a)] = 2*sin*a*omega /omega;" "6#/*&%#FrG\"\"\"7#,&-%\"HG6#,&%\"xGF&%\"aGF&F&-F*6#,&F-F&F.! \"\"F2F&*,\"\"#F&%$sinGF&F.F&%&omegaGF&F6F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 55 "Using the 2nd differentiation formula it follow s that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[-x^2* (H(x+a)-H(x-a))] = d^2/(d*omega^2);" "6#/*&%#FrG\"\"\"7#,$*&%\"xG\"\"# ,&-%\"HG6#,&F*F&%\"aGF&F&-F.6#,&F*F&F1!\"\"F5F&F5F&*&%\"dGF+*&F7F&*$%& omegaGF+F&F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[2*sin*a*omega/omega];" "6#7#*,\"\"#\"\"\"%$sinGF&%\"aGF&%&omegaGF&F)!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d/(d*omega)" "6#*&%\"dG\"\"\"*&F$F%%&omegaGF%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[2*omega^(-1)*sin*a*omega] = 2*a*omega^ (-1)*cos*a*omega-2*omega^(-2)*sin*a*omega;" "6#/7#*,\"\"#\"\"\")%&omeg aG,$F'!\"\"F'%$sinGF'%\"aGF'F)F',&*.F&F'F-F')F),$F'F+F'%$cosGF'F-F'F)F 'F'*,F&F')F),$F&F+F'F,F'F-F'F)F'F+" }{TEXT -1 3 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "so " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^ 2/(d*omega^2)" "6#*&%\"dG\"\"#*&F$\"\"\"*$%&omegaGF%F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[2*omega^(-1)*sin*a*omega] = ``;" "6#/7#*,\" \"#\"\"\")%&omegaG,$F'!\"\"F'%$sinGF'%\"aGF'F)F'%!G" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "-2*a^2*omega^(-1)* sin*a*omega-2*a*omega^(-2)*cos*a*omega-2*a*omega^(-2)*cos*a*omega+4*om ega^(-3)*sin*a*omega;" "6#,**.\"\"#\"\"\"*$%\"aGF%F&)%&omegaG,$F&!\"\" F&%$sinGF&F(F&F*F&F,*.F%F&F(F&)F*,$F%F,F&%$cosGF&F(F&F*F&F,*.F%F&F(F&) F*,$F%F,F&F1F&F(F&F*F&F,*,\"\"%F&)F*,$\"\"$F,F&F-F&F(F&F*F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -2*a^ 2*omega^(-1)*sin*a*omega-4*a*omega^(-2)*cos*a*omega+4*omega^(-3)*sin*a *omega;" "6#/%!G,(*.\"\"#\"\"\"*$%\"aGF'F()%&omegaG,$F(!\"\"F(%$sinGF( F*F(F,F(F.*.\"\"%F(F*F()F,,$F'F.F(%$cosGF(F*F(F,F(F.*,F1F()F,,$\"\"$F. F(F/F(F*F(F,F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Fr*[a^2*(H(x+a)-H( x-a))] = 2*a^2*omega^(-1)*sin*a*omega;" "6#/*&%#FrG\"\"\"7#*&%\"aG\"\" #,&-%\"HG6#,&%\"xGF&F)F&F&-F-6#,&F0F&F)!\"\"F4F&F&*.F*F&*$F)F*F&)%&ome gaG,$F&F4F&%$sinGF&F)F&F8F&" }{TEXT -1 18 ", it follows that " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[f(x)] = Fr*[(a^2 -x^2)*(H(x+a)-H(x-a))];" "6#/*&%#FrG\"\"\"7#-%\"fG6#%\"xGF&*&F%F&7#*&, &*$%\"aG\"\"#F&*$F+F2!\"\"F&,&-%\"HG6#,&F+F&F1F&F&-F76#,&F+F&F1F4F4F&F &" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -4*a*omega^(-2)*cos*a*omega+4*om ega^(-3)*sin*a*omega;" "6#/%!G,&*.\"\"%\"\"\"%\"aGF()%&omegaG,$\"\"#! \"\"F(%$cosGF(F)F(F+F(F.*,F'F()F+,$\"\"$F.F(%$sinGF(F)F(F+F(F(" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*``(sin*a*omega-a*omega*cos*a*ome ga)/(omega^3);" "6#/%!G*(\"\"%\"\"\"-F$6#,&*(%$sinGF'%\"aGF'%&omegaGF' F'*,F-F'F.F'%$cosGF'F-F'F.F'!\"\"F'*$F.\"\"$F1" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "alias(H=Heaviside):\n(a^2-x^2)*(H(x+a)-H(x-a));\n`Fourier transfor m`=inttrans[fourier](%,x,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& ,&*$)%#a|irG\"\"#\"\"\"F)*$)%\"xGF(F)!\"\"F),&-%\"HG6#,&F,F)F'F)F)-F06 #,&F,F)F'F-F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transfor mG,$*(\"\"%\"\"\"%&omegaG!\"$,&-%$sinG6#*&%#a|irGF(F)F(F(*(F0F(-%$cosG F.F(F)F(!\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "a = 2" "6#/%\"aG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F(omega) = 4*``(sin*2*omega-2*omega*cos*2*omega)/(o mega^3);" "6#/-%\"FG6#%&omegaG*(\"\"%\"\"\"-%!G6#,&*(%$sinGF*\"\"#F*F' F*F**,F1F*F'F*%$cosGF*F1F*F'F*!\"\"F**$F'\"\"$F4" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "F := w -> 4*(sin(2*w)-2*w*cos(2*w))/(w^3);\nplot(F(w),w=-8..8,col or=COLOR(RGB,.4,0,.9),title=`Fourier transform F(w) of f(x)`);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"wG6\"6$%)operatorG%&arrow GF(,$*(\"\"%\"\"\",&-%$sinG6#,$*&\"\"#F/9$F/F/F/*(F6F/F7F/-%$cosGF3F/! \"\"F/F7!\"$F/F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 583 242 242 {PLOTDATA 2 "6'-%'CURVESG6#7cq7$$!\")\"\"!$\"3cs(*y.>eu6!#=7$$!3OLLLLb C^w!#<$\"3I(yuR7r4H\"F-7$$!3?mmmOhzZtF1$\"3d,!Qw-;iq)!#>7$$!3LLLL`b`1q F1$!3P>1k%zC6j)!#?7$$!3#HLLL(G,jmF1$!3CC[510W87F-7$$!30nmm'*G7@jF1$!3O B+0XrS%)>F-7$$!3o+++5]jihF1$!3M%oW@Lej3#F-7$$!3XLLLBr9/gF1$!3MQd(*HfJ! )>F-7$$!3=mmm'4U+%eF1$!3)H!*e%HUqQ;F-7$$!3!)******pq$fn&F1$!3!Gw$>b#z6 2\"F-7$$!3fLLLj<]O`F1$\"3mM#oWyG*3kF97$$!3Q+++I]:)*\\F1$\"3]CN&[2N,_#F -7$$!3(QLL$G%RT#[F1$\"37JrnC**3tKF-7$$!3YmmmEQ7]YF1$\"3kcq_zLT?PF-7$$! 3;LLL3t[tXF1$\"3EXB2aIr#z$F-7$$!3))*******y]o\\%F1$\"3/uu]vM.yPF-7$$!3 emmmrU@?WF1$\"37$f2xbG1n$F-7$$!3HLLL`xdVVF1$\"3MB$y[esjY$F-7$$!3!ommmh G5<%F1$\"3/oq0;G;VEF-7$$!3I+++![z%)*RF1$\"3U**)pR*RKK8F-7$$!35++++U'>l $F1$!3&4zU)H$\\aV#F-7$$!3/+++?D.=LF1$!3zjyLn\\OSkF-7$$!3smmmT:TmJF1$!3 RN]Xt4y1zF-7$$!3SLLLj0z9IF1$!3yctK_Ng')))F-7$$!3)****\\7c<(pHF1$!3#yc` :KJ^0*F-7$$!3bmm;fXkCHF1$!3?2Nb#z$Qe\"*F-7$$!3cLL3d:dzGF1$!3or3mXjZ\"> *F-7$$!39+++b&)\\MGF1$!3!=q+m3m'\\\"*F-7$$!3smm\"HbD%*y#F1$!37i.yX)z$G !*F-7$$!3uLL$3b_Vu#F1$!33KCB))F-7$$!3I++v[&z#*p#F1$!3t2zo195I&)F-7 $$!3!pmmma1Ul#F1$!3;`!G(f*e]9)F-7$$!3/nmmmbZ,DF1$!3:\"eYch766'F-7$$!3= nmm'eW([BF1$!3Wf#\\gn12(GF-7$$!3yLLL[@3r@F1$\"3'[W*[7V4(\\#F-7$$!3S+++ 5(>M*>F1$\"3'\\;@%pC&3e*F-7$$!3\"RLL$[$eh$=F1$\"3`w\"*RL*)Q>i^p6F1$\"3Q?Vs-h]ieF17$$!3*)* *****Hn@05F1$\"3O+:^BJ?LpF17$$!3\"=LLL[uyL)F-$\"3eDoT4(e.)zF17$$!3qkmm m;eBmF-$\"30Hm'>d\"Q3*)F17$$!33mmm;dK\\]F-$\"3W5_wT>4#))e5F\\z7$$!3szmm;>)* z#*F9$\"3/&zhM\"o*H1\"F\\z7$$!3#R,+]ii[.&F9$\"3;E\"Q]Y&el5F\\z7$$!3?\" [LLLLu*yF?$\"3;De$f0Sm1\"F\\z7$$\"3MPmmTDu>OF9$\"3sh%e(Hx5m5F\\z7$$\"3 !GimmTG#H!)F9$\"3o#fqBa=R1\"F\\z7$$\"3$3mm\"H9(QC\"F-$\"3i1Fr&yz+1\"F \\z7$$\"3Qfmm;+#[o\"F-$\"3KO\"z&QUga5F\\z7$$\"3Ycmm\">/uV'*F17$$\"3x\"******p!R>lF-$\"3)[m!o#3O*f*)F17$$\"3)\\KLL)>lx\")F -$\"3-3xlWD'G2)F17$$\"3AemmmK\"f$)*F-$\"3q/7n$)fAqqF17$$\"3kKLLVf!\\: \"F1$\"3Ga(p,0B#ffF17$$\"3W******f0AE8F1$\"3*yp'\\b(f2#[F17$$\"3)))*** ***[=Q\\\"F1$\"3a$)yOI6lEPF17$$\"3M)*****>kTh;F1$\"3uVE#>#Qa$p#F17$$\" 3a)****\\.wN#=F1$\"3!R(oKr:J&y\"F17$$\"3u)*****\\ct&)>F1$\"3m$\\L$H3iC **F-7$$\"3m)****\\D'yl@F1$\"3uojDK*>Po#F-7$$\"3e)*****fo$eM#F1$!3)**Rx +%H\"oz#F-7$$\"3RlmmO.i2DF1$!3[21:zKj:iF-7$$\"3?KLL8QSpEF1$!3obcw2NQ&G )F-7$$\"3Ulmm6%)e7FF1$!3N,t!ov\\gi)F-7$$\"33******4IxbFF1$!3k*HENF$Q$) ))F-7$$\"3uKLL3w&*)z#F1$!3CUje\"yw31*F-7$$\"3&fmmm?U@%GF1$!3]J;oH$HA;* F-7$$\"3;*****\\!oK&)GF1$!3%znRWcC8>*F-7$$\"3#GLLLS6&GHF1$!3I>o'**e3A: *F-7$$\"3[mmm,gprHF1$!3-Ji3Hv1\\!*F-7$$\"3p*******f!)[,$F1$!39%Q=7!\\? ')))F-7$$\"3!GLL$esSrJF1$!3Mlyh7$Q^'yF-7$$\"3%fmmm\"R$zK$F1$!3.#eTP6kB L'F-7$$\"3s******zQ=qOF1$!305&=0s2L@#F-7$$\"3mJLLBW@#*RF1$\"3qI`%)y&\\ iF\"F-7$$\"3Mlmmw<_gTF1$\"3I7Q'e&)zod#F-7$$\"3.******H\"H)GVF1$\"3xR1Z vRs:MF-7$$\"3VKL$3t/6T%F1$\"3s$eS<#Q[^OF-7$$\"3%emm;L!Q$\\%F1$\"3)Hf%H g9@vPF-7$$\"3C****\\KflvXF1$\"3Wb^V=B$=z$F-7$$\"3mKLLL:$zl%F1$\"3skO!o lN%3PF-7$$\"3'fmm;R,-$[F1$\"37uAS#>G8D$F-7$$\"3E******\\7Z-]F1$\"3#f`C rP'f)\\#F-7$$\"32nmmYRIM`F1$\"3)*4_k<(**R`'F97$$\"3?mmm13ltcF1$!3GKr1; N\"=1\"F-7$$\"3k*****\\D>>%eF1$!3S!eK$\\:$Rk\"F-7$$\"33LLL.x=5gF1$!3uB *G3Dv$))>F-7$$\"3KmmmJ*3[;'F1$!3O%4\"=jF1$!3I )G9@#\\]')>F-7$$\"3?LLL8p&Qn'F1$!3;DQ&y2E/=\"F-7$$\"33mmmE/'3*pF1$!3-u br3ilx8F97$$\"3Q+++!H_)GtF1$\"3WMJNbj9*G)F97$$\"3O+++ION_wF1$\"3Ud2'*) [,;H\"F-7$$\"\")F*F+-%&TITLEG6#%?Fourier~transform~F(w)~of~f(x)G-%+AXE SLABELSG6$Q\"w6\"Q!F`\\m-%&COLORG6&%$RGBG$\"\"%!\"\"$F*F*$\"\"*Fh\\m-% %VIEWG6$;F(Ff[m%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 28 " be the function defined by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([a^2-(x^2-b)^2, abs(x) < sqrt(a+b)],[0, sqrt(a+b) <= x]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7 $,&*$%\"aG\"\"#\"\"\"*$,&*$F'F/F0%\"bG!\"\"F/F52-%$absG6#F'-%%sqrtG6#, &F.F0F4F07$\"\"!1-F;6#,&F.F0F4F0F'" }{TEXT -1 3 ", " }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{XPPEDIT 18 0 "0 < a+b;" "6#2\"\"!,&%\"aG\"\"\" %\"bGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 14 " for the case " }{XPPEDIT 18 0 "a = 4, b = 5 " "6$/%\"aG\"\"%/%\"bG\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "f := x -> piecewi se(x>-3 and x<3,16-(x^2-5)^2):\n'f(x)'=f(x);\nplot(f(x),x=-5..5,color= red,thickness=2,ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG-%*PIECEWISEG6$7$,&\"#;\"\"\"*$),&*$)F'\"\"#F.F.\"\"&!\"\"F 4F.F632,$F'F6\"\"$2F'F:7$\"\"!%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 485 268 268 {PLOTDATA 2 "6(-%'CURVESG6#7eq7$$!\"&\"\"!$F*F*7 $$!3YLLLe%G?y%!#f_PF/F+7$$!3K++vo1YZNF/F+7$$!3;LL3- OJNLF/F+7$$!3?mm\"zC!eHKF/F+7$$!3p***\\P*o%Q7$F/F+7$$!37m\"H#=qYpIF/F+ 7$$!3+L$3F9(3:IF/F+7$$!3))*\\(=nsqgHF/$\"3QxKKRV%)==F/7$$!3Kmmm\"RFj!H F/$\"3Ipu;.4**>TF/7$$!3!HL3xcH%eGF/$\"3Y0;+$Rrr%fF/7$$!3[***\\PuJ0\"GF /$\"31\\uE5[G&f(F/7$$!31m;z>RjiFF/$\"3mqFd9l!=2*F/7$$!33LL$e4OZr#F/$\" 36Ro[a&3%Q5!#;7$$!3UmmT&)G*og#F/$\"3C$=TasxuF\"Fio7$$!3u*****\\n\\!*\\ #F/$\"3YeZ`Ba$\\W\"Fio7$$!3y******\\Ow!R#F/$\"3)yL7\\2q([:Fio7$$!3%)** ***\\ixCG#F/$\"3o_Do=Cg&f\"Fio7$$!3C++]7#*QcAF/$\"3KGD*>dm\"*f\"Fio7$$ !3>++++3IIAF/$\"39kj'yX$f\"Fio7$$!3,+++]r%f7#F/$\"3G'pg6^Epd\"Fio7$ $!3#******\\KqP2#F/$\"3yE%GJCt5b\"Fio7$$!3am;a8s+z>F/$\"3)=:jm<'f#[\"F io7$$!39LL3-TC%)=F/$\"35-JQy*e)*Q\"Fio7$$!3#)**\\(ofh:x\"F/$\"3;41^'*e X`7Fio7$$!3[mmm\"4z)e;F/$\"3aRe'\\d&f%4\"Fio7$$!3Smm;HAUj:F/$\"3!y)y;D ;Mo%*F/7$$!3Mmmmm`'zY\"F/$\"3c)4l=)H`0zF/7$$!37L$3FMEpN\"F/$\"33(o;)[F HAgF/7$$!3#****\\(=t)eC\"F/$\"3GGR))fz\"H6%F/7$$!3OL$3x'*)fZ6F/$\"3[]: ^b')QNCF/7$$!3!ommmh5$\\5F/$\"3MI(*>'*\\5#)z!#=7$$!3tIL3xrs9%*F^v$!3!4 (f5=bW>#*F^v7$$!3S$***\\(=[jL)F^v$!3J[$>yB\"[LDF/7$$!3q%****\\Pw%4tF^v $!3)HxDXQ:E%RF/7$$!3)f***\\iXg#G'F^v$!3RT9BB\\o3_F/7$$!3$oK$3_:<6_F^v$ !3)4pGqh:\"ejF/7$$!3ndmmT&Q(RTF^v$!3m>Y;\\ci:tF/7$$!3Ihm\"HdGe:$F^v$!3 v_L#eF$*R,)F/7$$!3%\\mmTg=><#F^v$!3lhR(R=-0`)F/7$$!3FK$3Fpy7k\"F^v$!3Z *))Q)ygMJ()F/7$$!3g***\\7yQ16\"F^v$!3GJza1.!o())F/7$$!3iK$3_D)=`%)!#>$ !3/D6[oYfG*)F/7$$!3Epm\"zp))**z&Fdy$!3eK=^Z9Pm*)F/7$$!3#f+D19*yYJFdy$! 3i2jl)p)4!**)F/7$$!3vDMLLe*e$\\!#?$!3I\"y;*pjv***)F/7$$\"3+l;a)3RBE#Fd y$!3!)Hf!Q3#)[**)F/7$$\"3bsmTgxE=]Fdy$!3a``_IL#[(*)F/7$$\"37!o\"HKk>ux Fdy$!3SDz](R)fR*)F/7$$\"3womT5D,`5F^v$!3K\"HtoTR#*)))F/7$$\"3Gq;zW#)>/ ;F^v$!34$ykY1Y2%F^v$!3iB$y(>AJntF/7$$\"36QLe*[K56& F^v$!3e\"o!f*)R(fX'F/7$$\"3summ\"zXu9'F^v$!3c&*zQqwqj`F/7$$\"3#yLLe9i \"=sF^v$!3W,(H[&RFhSF/7$$\"3#4+++]y))G)F^v$!3J(4?v)R\\,EF/7$$\"3%>++Dc ljL*F^v$!38,+v#)*[I/\"F/7$$\"3H++]i_QQ5F/$\"3kSYx2^H)>'F^v7$$\"3U+](=- N(R6F/$\"3#*[xZ+)oDI#F/7$$\"3b++D\"y%3T7F/$\"3.\"ey+k;/.%F/7$$\"3G+]P4 kh`8F/$\"3U5pOEM`lfF/7$$\"3+++]P![hY\"F/$\"3kAny=K;vyF/7$$\"3KmmT5FEn: F/$\"38i!G0(36Fio7$$\"3cmm;z)Qjx\" F/$\"3%Q)p!ojP(f7Fio7$$\"3Y+++v.I%)=F/$\"37rAPd/#**Q\"Fio7$$\"3ML$ek`H @)>F/$\"3=D&4(*3h_[\"Fio7$$\"3?mm\"zpe*z?F/$\"35Uk8aJga:Fio7$$\"3&**\\ (oa_VL@F/$\"3#HmF(**))))z:Fio7$$\"3oL$e9\"=\"p=#F/$\"3An#*f'*HF&f\"Fio 7$$\"3c]P%)*3]O@#F/$\"3%f@=$G\\+*f\"Fio7$$\"3Vn\"H#o$)QSAF/$\"3CfLG'fi **f\"Fio7$$\"3I%e9mkErE#F/$\"3#)G$Rw#Q/)f\"Fio7$$\"3;,++D\\'QH#F/$\"3A /?\"GAXJf\"Fio7$$\"3/n;zp%*\\%R#F/$\"3gb)*=Q!zha\"Fio7$$\"3%HL$e9S8&\\ #F/$\"3/h>%H[n(\\9Fio7$$\"3%om;/6E.g#F/$\"3ojxXiFk*G\"Fio7$$\"3s++D1#= bq#F/$\"3SK$4/XR=1\"Fio7$$\"3yL3xc/%pv#F/$\"3grBGL>DO#*F/7$$\"3#om\"H2 FO3GF/$\"3kX\"3O1-em(F/7$$\"3')*\\7y&\\yfGF/$\"3#>>d<6kz*eF/7$$\"3\"HL L$3s?6HF/$\"3\"fi&H&[*[BRF/7$$\"3!)*\\i!R:/lHF/$\"3=mw " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 " H(x" "6#-%\"HG6#%\"xG" }{TEXT -1 36 " is the Heaviside function given by " }{XPPEDIT 18 0 "H(x)=PIECEWISE( [0 , x<0],[1 , x>0" "6#/-%\"HG6#%\"xG-%*PIECEWISEG6$7$\"\"!2F'F,7$\"\" \"2F,F'" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "c = sqrt(a+b)" "6#/%\"cG -%%sqrtG6#,&%\"aG\"\"\"%\"bGF*" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 14 " has the form " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f( x) = (a^2-(x^2-b)^2)*(H(x+c)-H(x-c))" "6#/-%\"fG6#%\"xG*&,&*$%\"aG\"\" #\"\"\"*$,&*$F'F,F-%\"bG!\"\"F,F2F-,&-%\"HG6#,&F'F-%\"cGF-F--F56#,&F'F -F8F2F2F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (a^2-b^2+2*b*x^2-x^4)* (H(x+c)-H(x-c));" "6#/%!G*&,**$%\"aG\"\"#\"\"\"*$%\"bGF)!\"\"*(F)F*F,F *%\"xGF)F**$F/\"\"%F-F*,&-%\"HG6#,&F/F*%\"cGF*F*-F46#,&F/F*F7F-F-F*" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[H(x+c)-H(x-c)] = 2*sin*c*ome ga/omega;" "6#/*&%#FrG\"\"\"7#,&-%\"HG6#,&%\"xGF&%\"cGF&F&-F*6#,&F-F&F .!\"\"F2F&*,\"\"#F&%$sinGF&F.F&%&omegaGF&F6F2" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Using the 2nd differentiation formula it follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Fr*[(a^2-b^2+2*b*x^2-x^4)*(H(x+c)-H(x-c ))] = ``;" "6#/*&%#FrG\"\"\"7#*&,**$%\"aG\"\"#F&*$%\"bGF,!\"\"*(F,F&F. F&%\"xGF,F&*$F1\"\"%F/F&,&-%\"HG6#,&F1F&%\"cGF&F&-F66#,&F1F&F9F/F/F&F& %!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "(a^2-b^2-2*b*D^2-D^4)*[2*sin*c*ome ga/omega];" "6#*&,**$%\"aG\"\"#\"\"\"*$%\"bGF'!\"\"*(F'F(F*F(%\"DGF'F+ *$F-\"\"%F+F(7#*,F'F(%$sinGF(%\"cGF(%&omegaGF(F4F+F(" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "D^2=d^2/(d*ome ga^2)" "6#/*$%\"DG\"\"#*&%\"dGF&*&F(\"\"\"*$%&omegaGF&F*!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "D^4=d^4/(d*omega^4)" "6#/*$%\"DG\"\"%*&% \"dGF&*&F(\"\"\"*$%&omegaGF&F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "G := w -> 2*sin(c*w)/w;\n(a^2-b^2)*G(w)-2*b*diff(G(w),w$2)-diff(G(w),w$4):\nsim plify(subs(c=sqrt(a+b),%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf *6#%\"wG6\"6$%)operatorG%&arrowGF(,$*&-%$sinG6#*&%\"cG\"\"\"9$F3F3F4! \"\"\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,*(-%$sinG6#*& -%%sqrtG6#,&%\"aG\"\"\"%\"bGF0F0%\"wGF0F0)F2\"\"#F0F1F0!\"#**-%$cosGF) F0F+F0)F2\"\"$F0F/F0F0*(F'F0F3F0F/F0!\"$*(F7F0F+F0F2F0!\"'F'\"\"'F0*$) F2\"\"&F0!\"\"!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Fr*[f(x)] = (16*omeg a^2*b*sin*c*omega-8*c*omega^3*a*cos*c*omega+24*omega^2*a*sin*c*omega+4 8*omega*c*cos*c*omega-48*sin*c*omega)/(omega^5);" "6#/*&%#FrG\"\"\"7#- %\"fG6#%\"xGF&*&,,*.\"#;F&*$%&omegaG\"\"#F&%\"bGF&%$sinGF&%\"cGF&F1F&F &*0\"\")F&F5F&F1\"\"$%\"aGF&%$cosGF&F5F&F1F&!\"\"*.\"#CF&*$F1F2F&F9F&F 4F&F5F&F1F&F&*.\"#[F&F1F&F5F&F:F&F5F&F1F&F&**F@F&F4F&F5F&F1F&F;F&*$F1 \"\"&F;" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "c=sqrt(a+b)" "6#/%\"cG-%%sqrtG6#,&%\"aG\"\"\"%\"bGF*" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 176 "alias(H=Heaviside):\n(a^2-(x^2-b)^2)*(H(x+sqr t(a+b))-H(x-sqrt(a+b)));\ninttrans[fourier](%,x,omega):\nsubs(sqrt(a+b )=c,simplify(%)):\n`Fourier transform`=expand(numer(%))/denom(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%#a|irG\"\"#\"\"\"F)*$),&*$)%\" xGF(F)F)%\"bG!\"\"F(F)F1F),&-%\"HG6#,&F/F)*$,&F'F)F0F)#F)F(F)F)-F46#,& F/F)F7F1F1F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Fourier~transformG* &,,*,\"\")\"\"\"%\"cGF)-%$cosG6#*&F*F)%&omegaGF)F))F/\"\"$F)%#a|irGF)! \"\"*&\"#[F)-%$sinGF-F)F3**F5F)F*F)F+F)F/F)F)**\"#;F)%\"bGF)F6F))F/\" \"#F)F)**\"#CF)F2F)F6F)F " 0 "" {MPLTEXT 1 0 22 "subs(\{a=4,b=5,c=3\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**( \"$w\"\"\"\"-%$sinG6#,$*&\"\"$F'%&omegaGF'F'F')F.\"\"#F'F'*&\"#[F'F(F' !\"\"*(\"$W\"F'-%$cosGF*F'F.F'F'*(\"#'*F'F6F')F.F-F'F3F'F.!\"&" }}} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "F(omega) = (176*ome ga^2*sin*3*omega-96*omega^3*cos*3*omega+144*omega*cos*3*omega-48*sin*3 *omega)/(omega^5);" "6#/-%\"FG6#%&omegaG*&,**,\"$w\"\"\"\"*$F'\"\"#F,% $sinGF,\"\"$F,F'F,F,*,\"#'*F,*$F'F0F,%$cosGF,F0F,F'F,!\"\"*,\"$W\"F,F' F,F4F,F0F,F'F,F,**\"#[F,F/F,F0F,F'F,F5F,*$F'\"\"&F5" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "F := w -> (176*w^2*sin(3*w)-96*w^3*cos(3*w)+144*w*cos(3*w)-48*s in(3*w))/(w^5);\nplot(F(w),w=-8..8,color=COLOR(RGB,.4,0,.9), title=`Fo urier transform F(w) of f(x)`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FGf*6#%\"wG6\"6$%)operatorG%&arrowGF(*&,**(\"$w\"\"\"\")9$\"\"#F0-%$ sinG6#,$*&\"\"$F0F2F0F0F0F0*(\"#'*F0)F2F9F0-%$cosGF6F0!\"\"*(\"$W\"F0F 2F0F=F0F0*&\"#[F0F4F0F?F0F2!\"&F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 591 269 269 {PLOTDATA 2 "6'-%'CURVESG6#7gv7$$!\")\"\"!$!3Q.'[\"*)*>KJ* !#=7$$!3OLLLLbC^w!#<$\"3-`ez/dQ?fF-7$$!3A+++N3_*\\(F1$\"3iY@I-ME]7F17$ $!3?mmmOhzZtF1$\"3K*3Hcr&H.i]&F1$\"3eM:U&ojkJ\"F17$$!3fLLLj<]O`F1$\"3n!) p.Vsp3FF17$$!3')******z]\">D&F1$\"3UT()4!)e'*HKF17$$!3)pmmmRGt;&F1$\"3 3u8s$p`)zNF17$$!3c+++b].D^F1$\"3k%*=[-M!)zOF17$$!39MLL8c%F17$$!35++++U'>l$F1$!3g%>b^k?M G$F17$$!33+++g$)*\\[$F1$\"3)GMp2g![@!)!#>7$$!3/+++?D.=LF1$\"3fo())3:) \\GTF17$$!3smmmT:TmJF1$\"3rW[]5!#; 7$$!3)****\\7c<(pHF1$\"3nCUIcn.26F`z7$$!3bmm;fXkCHF1$\"3#RD>`Xy*[6F`z7 $$!3cLL3d:dzGF1$\"3cJ4**\\#=X<\"F`z7$$!39+++b&)\\MGF1$\"3s-\"p2`3B=\"F `z7$$!3@L$eR0i>\"GF1$\"3'>\")e<:v\"z6F`z7$$!3smm\"HbD%*y#F1$\"3+eRvV>= r6F`z7$$!3C+](=0*)ow#F1$\"37q83WZ@e6F`z7$$!3uLL$3b_Vu#F1$\"3&=lvjFt,9 \"F`z7$$!3I++v[&z#*p#F1$\"3;6\"\\h=R&)3\"F`z7$$!3!pmmma1Ul#F1$\"3q:,8Z ax:5F`z7$$!3/nmmmbZ,DF1$\"34Te>kv\"*3hF17$$!3=nmm'eW([BF1$!3?:%)3!=vaL #F-7$$!3G++]nL\"*fAF1$!3o8\"*e]xC'z%F17$$!3yLLL[@3r@F1$!3Q=][#y8Y#)*F1 7$$!3Knm;H4D#3#F1$!3tO,F^zr8:F`z7$$!3S+++5(>M*>F1$!3G]?*p'Gx_?F`z7$$!3 ;nm;H!*y9>F1$!3-:6q9,j=DF`z7$$!3\"RLL$[$eh$=F1$!3%\\ZUkost&HF`z7$$!3o+ +]nw_d(Hr,D\")RF`z7$$!3aLLL[jN1:F1$!3?t\\cLvJqTF`z7$$!3wm;ao()y %[\"F1$!3q2wZ0=\\+UF`z7$$!3'****\\()=@KY\"F1$!3#y&f)oJFNA%F`z7$$!3;L$e *3OlT9F1$!3mN(R!eDIRUF`z7$$!3gmm;Hg3?9F1$!3?.%Qoe8xC%F`z7$$!3-+]P\\%=& )R\"F1$!3)3_\"RbWn[UF`z7$$!3CLLep3&pP\"F1$!3X#pK&)e<@C%F`z7$$!3Wm;z*G$ Qb8F1$!3wT7)G6%*zA%F`z7$$!3l******4d\"QL\"F1$!3$)Qv=NTF1UF`z7$$!3s**** *\\'fm^7F1$!3]0%yxjBS0%F`z7$$!3w******>i^p6F1$!3)eAla'Gs$z$F`z7$$!3%)* ****\\Zmt3\"F1$!3(f)pbj*e9V$F`z7$$!3*)******Hn@05F1$!31O+kOTFxHF`z7$$! 3Olmm\"*3-&>*F-$!3gON.mLC?CF`z7$$!3\"=LLL[uyL)F-$!3!Q)*zX3Ztz\"F`z7$$! 3E)****\\2G2[(F-$!3S0OH\\@\"48\"F`z7$$!3qkmmm;eBmF-$!3m_%Hv2JWX%F17$$! 3Rlmm\"p`k$eF-$\"3?b/WMG#zy\"F17$$!33mmm;dK\\]F-$\"3%)\\P*oS\")Rx(F17$ $!3wmmmTx>iUF-$\"3![d2Tes4L\"F`z7$$!3Wnmmm(p]Z$F-$\"3Y*ehYW@9#=F`z7$$! 3HMLL3f/EEF-$\"3!yTjQBg6E#F`z7$$!37,++]?-x)*z#*F`y$\"3SO!>np(e+GF`z7$$!3?\"[LLLLu*y!#?$\"3-=OF`y$\"33lWyWz(y'GF`z7 $$\"32Im;za[CeF`y$\"3A8Y*[RV'[GF`z7$$\"3!GimmTG#H!)F`y$\"3(o#f[&\\$\\? 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